The present invention is an improvement upon the computer-aided techniques of my prior U.S. Pat. No. 5,606,144, and earlier MIT doctoral thesis Musical Variations from a Chaotic Mapping, and papers “Musical Variations from a Chaotic Mapping” (Chaos, 1996), “A chaotic mapping for music and image variations” (Proc. Fourth Int'l. Chaos Conference, 1998), and “Creating Musical Variation” (Science, 2008). The invention is a method for extending the variation capabilities of musical compositions, including both printed and recorded versions of the same. It should be noted that the term “variation” is used herein to refer to any process by which some features of a work (such as a musical piece) change while other features remain the same, and the term “variation technique” is used herein to refer to any modification and/or compositional procedure.
The present invention is mainly described in this paper in terms of its application to musical symbols or notes, following the illustrations of my said prior patent. However, it should be noted that the present invention is not limited to variations of music, and that embodiments of the present invention are generically applicable to all types of symbols, characters, images, and such like.
In my prior patent, a chaotic mapping technique is provided for generating, for example, musical variations of a given work. This technique, based on the sensitivity of chaotic trajectories to initial conditions, produces changes in the pitch sequence of a piece. The mapping in embodiments of my prior patent utilize two chaotic trajectories from the Lorenz equations—a system comprising three nonlinear first order differential equationsdx/dt=σ(y−x)  (1)dy/dt=rx−y−xz  (2)dz/dt=xy−bz,  (3)where σ=10, b=8/3, and r=28 (E. N. Lorenz, “Deterministic nonperiodic flow,” J. Atmos. Sci. 20, 130-141 (1963)). Other embodiments use other chaotic systems.
In the original chaotic mapping of my prior patent, the x-components {xi} of a reference chaotic trajectory are paired with a sequence of musical pitches {pi}. Each pi is then marked on the x axis at the point designated by its xi. In this way, the x-axis becomes a pitch axis configured according to the notes of the source composition. Then, a second chaotic trajectory, whose initial condition differs from the first, is launched. Its x-components trigger pitches on the pitch axis that vary in sequence from the source work, thus creating a variation. The possible variations are virtually infinite in number, and can be very close to the source, diverge substantially, or achieve degrees of variability in between these two extremes.
The Lorenz equations arise in applications ranging from lasers to private communications, and have also served as generators of “chaotic” music, where a chaotic system is allowed to free-run and its output converted into a series of notes, rhythms, and other musical attributes in order to create a piece from scratch. However, these approaches did not generate variations on an already completed piece.
The original chaotic mapping of my said U.S. Pat. No. 5,606,144 operates on the note list of a source piece, e.g., one provided by a MIDI file score, and treats each successive pitch, or simultaneous sounding of pitches (chord), as a parsed event. This represents the default parsing of the MIDI file score.
However, despite its ability to produce variations which can be delightful, appealing to musicians and non-musicians alike, the original chaotic mapping described in my previous patent has limitations. For instance, it requires that no note occur in the variation that was not already present in the input source file. While this can create a ‘pitch space’ link between the variation and the input source file, i.e., the variation is composed of pitches found in the input file, a richer variation technique would allow pitches outside the pitch space of the input file, if desired. Furthermore, my original chaotic mapping technique does not offer choices to apply different variation techniques within one piece.
Also, my original chaotic mapping approach applies only to a discrete-time signal representation of a source piece, e.g., a MIDI file score, prose, poetry, image, etc., but not to a continuous-time signal representation of a work, e.g., an audio recording, video, film, spoken word art, video games, and so on.
What is needed, therefore, is an improved chaotic mapping method which can introduce notes and/or symbols into a variation which are outside the pitch space of the input file, which can combine and offer choices between both new and time-honored variation and/or compositional procedures, and can generate variations not only of a discrete-time signal representation but also of a continuous-time signal representation of an existing work.